| L(s) = 1 | + 0.561·2-s − 7.68·4-s − 7·7-s − 8.80·8-s + 25.8·11-s − 15.5·13-s − 3.93·14-s + 56.5·16-s + 95.1·17-s − 143.·19-s + 14.4·22-s − 77.5·23-s − 8.73·26-s + 53.7·28-s + 204.·29-s − 40.6·31-s + 102.·32-s + 53.4·34-s + 95.6·37-s − 80.5·38-s − 24.7·41-s + 282.·43-s − 198.·44-s − 43.5·46-s − 257.·47-s + 49·49-s + 119.·52-s + ⋯ |
| L(s) = 1 | + 0.198·2-s − 0.960·4-s − 0.377·7-s − 0.389·8-s + 0.707·11-s − 0.331·13-s − 0.0750·14-s + 0.883·16-s + 1.35·17-s − 1.73·19-s + 0.140·22-s − 0.702·23-s − 0.0658·26-s + 0.363·28-s + 1.30·29-s − 0.235·31-s + 0.564·32-s + 0.269·34-s + 0.425·37-s − 0.343·38-s − 0.0941·41-s + 1.00·43-s − 0.679·44-s − 0.139·46-s − 0.798·47-s + 0.142·49-s + 0.318·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 0.561T + 8T^{2} \) |
| 11 | \( 1 - 25.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 95.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 77.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 95.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 24.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 282.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 257.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 257.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 651.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 451.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 832.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 174.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 47.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.49e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.36e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.522159381079869159984886151380, −8.170915117258542673441349737224, −6.94308381369935715011788674764, −6.13648521869361046893741555063, −5.34151541851832069222826536003, −4.31721906385571152202760831783, −3.75219163088503200248260963520, −2.61079507159791105883210867362, −1.15656329597800068966987410115, 0,
1.15656329597800068966987410115, 2.61079507159791105883210867362, 3.75219163088503200248260963520, 4.31721906385571152202760831783, 5.34151541851832069222826536003, 6.13648521869361046893741555063, 6.94308381369935715011788674764, 8.170915117258542673441349737224, 8.522159381079869159984886151380
